The form of the current density has been derived for all of the usual models of heterostructure electronic states. While one can usually find a way to obtain correct answers in a manual calculation without knowledge of the general expressions for J presented here, they provide a useful check on the results. The use of these expressions becomes more necessary if one seeks to develop the machinery for automatic computations (numerical or symbolic) applicable to a wide variety of heterostructures.
The reader will have noticed that, in conformance to the practice in quantum-mechanics texts, expressions for the expectation values of J have been presented, not expressions for the operator itself. It is very difficult to represent J as an ordinary quantum-mechanical operator. Actually, J is much more naturally expressed as a superoperator which acts upon a density operator [20]. All of the results presented here may be derived much more elegantly in a superoperator formalism, at the cost of an unfamiliar notation.